Problem 10 - Algebra 2 and Trigonometry

Question

The lengths of the sides of \(\triangle A B C\) are given. For each triangle, \(\angle C\) is the right angle and \(\mathrm{m} \angle A<\mathrm{m} \angle B .\) Find: a. \(\sin A\) b. \(\cos A\) c. \(\tan A\). \(6,3 \sqrt{5}, 9\)

Step-by-Step Solution

Answer

\( \sin A = \frac{2}{3}, \cos A = \frac{\sqrt{5}}{3}, \tan A = \frac{2\sqrt{5}}{5}. \)

Step 1: Understand the Triangle Configuration

Given a right triangle \( \triangle ABC \) with \( \angle C = 90^\circ \), \( \angle A < \angle B \), and side lengths given as \( 6, 3\sqrt{5}, \) and \( 9 \). We need to identify which side corresponds to each length: the hypotenuse, opposite, and adjacent sides relative to angle \( A \).

Step 2: Assign Side Lengths to Triangle Sides

In a right triangle, the longest side, \( 9 \), is the hypotenuse. Since \( \angle A < \angle B \), the shorter side \( 6 \) is opposite \( \angle A \), and the remaining side \( 3\sqrt{5} \) is adjacent to \( \angle A \).

Step 3: Calculate \( \sin A \)

The sine of angle \( A \) is defined as the ratio of the opposite side to the hypotenuse. Thus, \[ \sin A = \frac{6}{9} = \frac{2}{3}. \]

Step 4: Calculate \( \cos A \)

The cosine of angle \( A \) is the ratio of the adjacent side to the hypotenuse. Therefore, \[ \cos A = \frac{3\sqrt{5}}{9} = \frac{\sqrt{5}}{3}. \]

Step 5: Calculate \( \tan A \)

The tangent of angle \( A \) is the ratio of the opposite side to the adjacent side. Thus, \[ \tan A = \frac{6}{3\sqrt{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}. \]

Key Concepts

Right Triangle

A right triangle is a special type of triangle where one of the angles is exactly 90 degrees, known as a right angle. This angle is usually denoted by the letter C in a triangle labeled ABC. Right triangles have distinct properties that make them unique and very useful in trigonometry.

These triangles consist of three sides:

• The hypotenuse is the longest side and is always opposite the right angle.
• The opposite side is across from a given angle of interest, excluding the right angle.
• The adjacent side is next to the angle of interest and is not the hypotenuse.

Understanding these sides is crucial for solving problems related to right triangles, as the trigonometric functions sine, cosine, and tangent involve these specific side lengths.

Sine

The sine function is a fundamental trigonometric function that is particularly useful in right triangles. It relates the angle of interest in a right triangle to the ratio of specific side lengths. For an angle A in a right triangle, sine is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

In mathematical terms, this relationship is expressed as:

• \(\sin A = \frac{\text{Opposite side}}{\text{Hypotenuse}}\)

In the example given, for angle A, the opposite side measures 6 units and the hypotenuse is 9 units. Thus, the sine of angle A can be calculated as \( \sin A = \frac{6}{9} = \frac{2}{3} \).

This ratio remains constant for any angle A within right triangles, determined by the ratio of these specific sides.

Cosine

The cosine function is another key concept in trigonometry that describes the relationship between an angle in a right triangle and the lengths of its sides. Specifically, cosine relates the angle of interest to the adjacent side and the hypotenuse.

Mathematically, cosine is defined as:

• \(\cos A = \frac{\text{Adjacent side}}{\text{Hypotenuse}}\)

Using our earlier example, where the adjacent side to angle A is \( 3\sqrt{5} \) and the hypotenuse is 9, the cosine of angle A is calculated by \( \cos A = \frac{3\sqrt{5}}{9} = \frac{\sqrt{5}}{3} \).

Cosine is particularly useful for calculating distances and understanding angles in physics and engineering, providing insights into the spatial relationships within right triangles.

Tangent

Tangent is a trigonometric function that provides insight into the relationship between the two non-hypotenuse sides of a right triangle. In terms of angle A, the tangent function is the ratio of the opposite side to the adjacent side.

The tangent function is represented as:

• \(\tan A = \frac{\text{Opposite side}}{\text{Adjacent side}}\)

Referring back to the triangle example, the opposite side of angle A measures 6 units, while the adjacent side measures \( 3\sqrt{5} \). Therefore, the tangent of angle A is \( \tan A = \frac{6}{3\sqrt{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5} \) after rationalizing.

Tangent is useful in various fields such as navigation and computer graphics, offering ways to compute unmeasured distances and heights efficiently.